Related papers: A combinatorial Yamabe problem on two and three di…

We consider the case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular…

The mixed scalar curvature is one of the simplest curvature invariants of a foliated Riemannian manifold. We explore the problem of prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the…

This is the second paper of our series of papers on one dimensional conformal metric flows. In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in math.AP/0611254. We prove the global…

For a closed Riemannian manifold of dimension $n\geq 3$ and a subgroup $G$ of the isometry group, we define and study the $G-$equivariant second Yamabe constant and we obtain some results on the existence of $G-$invariant nodal solutions of…

In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the $(-1,-1,-1)$ type generalized circle packing metric introduced by Guo-Luo \cite{GL2}. To find…

We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble. As an application of our result, we…

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity…

In this paper we propose a novel second-order accurate well balanced scheme for shallow water equations in general covariant coordinates over manifolds. In our approach, once the gravitational field is defined for the specific case, one…

In this paper, for a given compact 3-manifold with an initial Riemannian metric and a symmetric tensor, we establish the short-time existence and uniqueness theorem for extension of cross curvature flow. We give an example of this flow on…

We consider the problem of prescribing conformally the scalar curvature on compact manifolds of positive Yamabe class in dimension $n \geq 5$. We prove new existence results using Morse theory and some analysis on blowing-up solutions,…

In this note, we survey recent advances in the study of dynamical properties of the space of surfaces with constant curvature in three-dimensional manifolds of negative sectional curvature. We interpret this space as a two-dimensional…

For triangulated surfaces and any $p>1$, we introduce the combinatorial $p$-th Calabi flow which precisely equals the combinatorial Calabi flows first introduced in H. Ge's thesis when $p=2$. The difficulties for the generalizations come…

We consider a parabolic-like systems of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds. We find that in the two-dimensional case there is…

We introduce the weighted Yamabe flow $\frac{\partial g}{\partial t}=(r^m_{\phi}-R^m_{\phi})g$, $\frac{\partial \phi}{\partial t}=\frac{m}{2}(R^m_{\phi}-r^m_{\phi})$ on a smooth metric measure space $(M^n, g, e^{-\phi}{\rm dvol}_g, m)$,…

We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open…

In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross…

Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial…

Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.

We study Yamabe metrics, and the moduli space of Yamabe metrics, on an arbitrary closed 3-manifold M. The main focus is on the boundary behavior of the moduli space, i.e. the behavior of degenerating sequences of unit volume Yamabe metrics…

In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. In three-dimensional manifold, we prove a minimal volume growth, an estimate of integral…